Problem

Source: China Western Mathematical Olympiad 2008

Tags: induction, ceiling function, logarithms, algebra proposed, algebra



Given an integer $ m\geq 2$, and two real numbers $ a,b$ with $ a > 0$ and $ b\neq 0$. The sequence $ \{x_n\}$ is such that $ x_1 = b$ and $ x_{n + 1} = ax^{m}_{n} + b$, $ n = 1,2,...$. Prove that (1)when $ b < 0$ and m is even, the sequence is bounded if and only if $ ab^{m - 1}\geq - 2$; (2)when $ b < 0$ and m is odd, or when $ b > 0$ the sequence is bounded if and only if $ ab^{m - 1}\geq\frac {(m - 1)^{m - 1}}{m^m}$.